A114289 -id:A114289 - cp's OEIS frontend

A114290 Number of oriented n-dimensional polytopes with n+3 vertices, meaning that two polytopes are identified if they have the same combinatorial type and there exists an orientation-preserving homeomorphism mapping the first polytope to the second polytope.

0, 1, 7, 38, 170, 617, 1979, 5859, 16571, 45516, 123159, 330736, 885780, 2372305, 6362965, 17102719, 46078541, 124440388, 336829857, 913658780, 2483217288, 6761405513, 18441239903, 50375429081, 137807555515, 377492301876

Offset: 1

Éric Fusy (eric.fusy(AT)inria.fr), Nov 21 2005

nonn

B. Grünbaum, Convex Polytopes, Springer-Verlag, 2003, Second edition prepared by V. Kaibel, V. Klee and G. M. Ziegler, p. 121a.

Éric Fusy, Counting d-polytopes with d+3 vertices, arXiv:math/0511466 [math.CO], 2005.
Éric Fusy, Counting d-polytopes with d+3 vertices, Electron. J. Comb. 13 (2006), no. 1, research paper R23, 25 pp.
E. K. Lloyd, The number of d-polytopes with d+3 vertices, Mathematika 17 (1970), 120-132.

Cf. A000943, A114289, A114291.

N:=30: with(numtheory): G:=-ln(1-2*x^3/(1-2*x)^2): H:=-log(1-2*x)+ln(1-x): K:=-(x^10+3*x^9-3*x^8-7*x^7+4*x^6+4*x^5+4*x^4+3*x^3-2*x^2+1)*x/(1-x)^5/(x+1)^3: series(1/(x^3-x^4)*(1/2*sum(phi(2*r+1)/(2*r+1)*subs(x=x^(2*r+1),G),r=0..N)+sum(phi(r)/r*subs(x=x^r,H),r=1..N)+K),x,N);

terms = 26;
G[x_] = -Log[1 - 2 (x^3/(1 - 2 x)^2)];
H[x_] = -Log[1 - 2 x] + Log[1 - x];
K[x_] = -(x^10 + 3 x^9 - 3 x^8 - 7 x^7 + 4 x^6 + 4 x^5 + 4 x^4 + 3 x^3 - 2 x^2 + 1) x/(1 - x)^5/(x + 1)^3;
1/(x^3 - x^4) (1/2 Sum[EulerPhi[2 r + 1]/(2 r + 1) G[x^(2 r + 1)], {r, 0, terms+3}] + Sum[EulerPhi[r]/r H[x^r], {r, 1, terms+3}] + K[x]) + O[x]^(terms+2) // CoefficientList[#, x]& // Rest // Most // Round (* Jean-François Alcover, Dec 14 2018 *)

A114291 Number of combinatorial types of achiral n-dimensional polytopes with n+3 vertices, where a polytope is achiral if one of its geometric realizations has a reflection-symmetry.

Original entry on oeis.org

0, 1, 7, 24, 62, 141, 287, 561, 1035, 1886, 3319, 5838, 10030, 17323, 29395, 50291, 84795, 144374, 242641, 412126, 691522, 1173151, 1966929, 3334931, 5589311, 9474106, 15875699, 26906538, 45083426, 76404103, 128014623, 216944163

Offset: 1

Éric Fusy (eric.fusy(AT)inria.fr), Nov 21 2005

nonn

B. Grünbaum, Convex Polytopes, Springer-Verlag, 2003, Second edition prepared by V. Kaibel, V. Klee and G. M. Ziegler, p. 121a.

Éric Fusy, Counting d-polytopes with d+3 vertices, arXiv:math/0511466 [math.CO], 2005.
Éric Fusy, Counting d-polytopes with d+3 vertices, Electron. J. Comb. 13 (2006), no. 1, research paper R23, 25 pp.
E. K. Lloyd, The number of d-polytopes with d+3 vertices, Mathematika 17 (1970), 120-132.
Index entries for linear recurrences with constant coefficients, signature (2, 6, -14, -12, 38, 8, -54, 5, 44, -12, -20, 8, 4, -2).

Cf. A000943, A114289, A114290.

LinearRecurrence[{2, 6, -14, -12, 38, 8, -54, 5, 44, -12, -20, 8, 4, -2}, {0, 1, 7, 24, 62, 141, 287, 561, 1035, 1886, 3319, 5838, 10030, 17323}, 32] (* Jean-François Alcover, Dec 14 2018 *)

concat(0, Vec((2*x^11+4*x^10-2*x^9-15*x^8-5*x^7+23*x^6+15*x^5 -17*x^4-14*x^3+4*x^2 +5*x+1)*x^2/ (-1+x)^5/(2*x^6-4*x^4+4*x^2-1)/(x+1)^3 + O(x^50))) \\ Michel Marcus, Dec 12 2014

G.f.: (2*x^11+4*x^10-2*x^9-15*x^8-5*x^7+23*x^6+15*x^5-17*x^4 -14*x^3 +4*x^2+5*x+1) *x^2 / ((-1+x)^5*(2*x^6-4*x^4+4*x^2-1)*(x+1)^3).

cp's OEIS Frontend

A114290 Number of oriented n-dimensional polytopes with n+3 vertices, meaning that two polytopes are identified if they have the same combinatorial type and there exists an orientation-preserving homeomorphism mapping the first polytope to the second polytope.

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